Investment Course - 2005 Day Three: Fixed-Income Analysis and Portfolio Strategies 3-1 The Role of Fixed-Income Securities in the Financial Markets and Portfolio Management 3-2 U.S. & Chilean Yield Curves: Feb 2004 Feb 2005 3-3 U.S. Yield Curve and Credit Spreads: February 2005 3-4 Historical Data on U.S. Credit Spreads: Rating-Class Average Yields Year 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988
1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 Treasury Aaa-rated 7.89% 8.73% 8.74 9.63 10.81 11.94 12.87 14.17 12.23 13.79 10.84 12.04 11.99 12.71 10.75 11.37
8.14 9.02 8.64 9.38 8.98 9.71 8.58 9.26 8.74 9.32 8.16 8.77 7.52 8.14 6.45 7.22 7.41 7.97 6.93 7.59 6.80 7.37 6.67 7.27 5.69 6.53 6.14 7.05 6.23 7.04 5.70 6.51 5.43
6.11 4.96 5.66 Aa-rated 8.92% 9.94 12.50 14.75 14.41 12.42 13.31 11.82 9.47 9.68 9.94 9.46 9.56 9.05 8.46 7.40 8.15 7.72 7.55 7.48 6.80 7.36 7.18 6.65 6.27 6.11 A-rated
9.12% 10.20 12.89 15.29 15.43 13.10 13.74 12.28 9.95 9.99 10.24 9.74 9.82 9.30 8.62 7.58 8.28 7.83 7.69 7.54 6.93 7.53 7.46 6.99 6.52 6.38 Baa-rated 9.49% 10.69 13.67 16.04 16.11
13.55 14.19 12.72 10.39 10.58 10.83 10.18 10.36 9.80 8.98 7.93 8.63 8.20 8.05 7.87 7.22 7.88 8.10 7.73 7.11 6.76 3-5 Historical Data on U.S. Credit Spreads: Spreads over Treasury Year 1978 1979 1980 1981 1982 1983 1984
1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 TSY 7.89% 8.74 10.81 12.87 12.23 10.84 11.99 10.75 8.14 8.64 8.98 8.58
8.74 8.16 7.52 6.45 7.41 6.93 6.80 6.67 5.69 6.14 6.23 5.70 5.43 4.96 Aaa - T 84 bp 89 113 130 156 120 72 62 88 74 73 68 58 61 62 77 56
66 57 60 84 91 81 81 68 70 Aa - T 103 bp 120 169 188 218 158 132 107 133 104 96 88 82 89 94 95 74 79 75 81 111 122
95 95 84 115 A-T 123 bp 146 208 242 320 226 175 153 181 135 126 116 108 114 110 113 87 90 89 87 124 139 123 129 109 142
Baa - T 160 bp 195 286 317 388 271 220 197 225 194 185 160 162 164 146 148 122 127 125 120 153 174 187 203 168 180 3-6 Latin American Long-Term Credit Ratings: February 2005 3-7
Par vs. Spot Yield Curves Generally speaking, bonds are considered capital market instruments in that they have maturities greater than one year. Another important way in which bonds differ from more basic short-term money market securities (such as T-bills or Commercial Paper) is that they can provide both capital gains and periodic interest to the investor. Thus, the bond's yield-to-maturity reflects the expected annual return to the investment, taking into account both sources of payoff throughout the remainder of the bond's life. A yield curve is a graphical representation of the investment possibilities in the bond market at a particular moment in time. Specifically, the yield curve depicts the yields-to-maturity available to investors on a series of bonds that are alike in all respects (e.g., default risk, liquidity, tax considerations) except maturity. Yield-toMaturity 10% 9% 8% 1 yr 2 yr 3 yr Maturity 3-8 Par vs. Spot Yield Curves (Cont.) A par value yield curve summarizes the yields for coupon-bearing
instruments where the coupon rate is equal to the yield-to-maturity. Assuming that the above example is based a collection of Eurobonds (i.e., bonds that pay an annual coupon), the 10% yield for the three-year instrument can be interpreted as the average annual return that the investor can expect if he or she: (i) Holds the bond until maturity, (ii) Reinvests all intermediate cash flows (i.e., the first two coupons) at the same 10% rate for the remaining time until maturity. A spot, or zero coupon, yield curve summarizes the yields for non-couponbearing instruments (i.e., pure discount bonds). These yields can therefore be interpreted as more of a "pure" return since there is no concern about having to reinvest intermediate coupon cash flows. For example, if the above yield curve corresponded to zero coupon securities, the 10%, threeyear yield would represent the average annual price appreciation in the bond if it were held to maturity. 3-9 U.S. Par and Spot Yield Curves: February 2005 3 - 10 Par vs. Spot Yield Curves (cont.) In general, there is no specific shape that the yield curve must take. That is, at any point in time the yield curve can be upward-sloping, flat, or downward-sloping. There are several classical theories that purport to explain yield curve shapes: (i) Unbiased Expectations, which suggests that the presence of intertemporal arbitrage forces securities of all maturities to reflect the consensus beliefs of the investing public as to future economic conditions; (ii) Segmented Markets, which suggests that yields for various maturities are determined by the supply and demand characteristics of investors in a specific market, with no regard to what happens in the markets for securities with different maturities.
(iii) Liquidity Preference, which essentially says that yields are determined by unbiased expectations but with the addition of a premium to the longer maturity instruments to reflect the additional default exposure; and (iv) Preferred Habitat, which is a more general version of the liquidity preference hypothesis. It assumes that borrowers and lenders alike have a natural maturity for securities based on institutional characteristics and will venture outside of that range to attain a higher expected rate of return (or lower cost of funds) only if the expected gain outweighs the risk. If in each maturity range the demand for funds equals the supply, then the level of the interest rate depends on expectations for future rates (so that the expectations theory holds). If not, the rate will adjust to bring the market into equilibrium such that no further maturity shifting is desirable. Empirical evidence provides some evidence to support all four theories, with the unbiased expectations hypothesis working best for short-term securities. 3 - 11 Implied Spot Yield Curves The implied zero-coupon (or "strip") yield curve is the sequence of zero-coupon (or pure discount) interest rates satisfying the no-arbitrage condition that both coupon-stripping and bond reconstitution exactly break even. Coupon-stripping is a bond arbitrage strategy whereby the dealer buys a coupon-bearing bond and then sells the coupons and principal separately as zero-coupon securities. Profit obtains when the total sale price of the zeros exceeds the purchase price of the coupon bond. Bond reconstitution is a bond arbitrage strategy whereby the dealer buys zero-coupon bonds in sufficient amounts and with appropriate maturity dates to constitute the exact same cash flows as on a particular coupon bond. Profit obtains when the sale price of the coupon bonds exceeds the total purchase price of the zeros. Example: Assume semiannual coupon payments on the following bonds. Maturity 0.5 1.0 1.5
2.0 Coupon Rate 3.00% 3.25% 4.00% 4.00% Price Yield-to-Maturity 99.926 99.795 100.318 100.000 3.15% (s.a.) 3.46% (s.a.) 3.78% (s.a.) 4.00% (s.a.) Notice that the 0.5-year security already is a zero-coupon bond since there is only one coupon remaining. 3 - 12 Implied Spot Yield Curves (cont.) First, diagram the cash flows on the 0.5-year and 1.0-year bonds: 101.5 101.625 1.625
0.5 0.5 99.926 1.0 99.795 Suppose that the 1.0-year bond is bought for 99.795. The first coupon of 1.625 can be sold as a 0.5-year zero-coupon security priced at 1.600 to yield 3.15% (s.a.) [i.e., (1.625) (1.01575), where 1.575% is the periodic equivalent of the nominal annual yield of 3.15%]. If coupon-stripping were to break even, the 1.0-year cash flow of 101.625 must then sell for 98.195, calculated as the difference between the purchase price of 99.795 and the receipt of 1.600. The implied 1.0-year zero-coupon rate is therefore 3.463% (s.a.), which is founding by solving 98.195 = (101.625) (1 + Z2/2)2 for Z2. Second, diagram the 1.5-year bond: 102 2 2 0.5 1.0 1.5 100.318 The 1.5-year bond can be bought for 100.318. Following the process established above, the first two coupons of 2 can be sold as a 0.5-year and 1.0-year zeros priced at 1.969 and 1.932 to yield
3.15% (s.a.) and 3.463% (s.a.), respectively. If coupon-stripping were then to break even, the 1.5year cash flow of 102 must sell for 96.417 [= 100.318 - 1.969 - 1.932]. The 1.5-year implied zerocoupon rate therefore must be 3.788% (s.a.) 3 - 13 Implied Spot Yield Curves (cont.) Next, diagram the 2.0-year security: 102 2 2 2 0.5 1.0 1.5 2.0 100 Consider a dealer reconstituting this bond and selling it for par value of 100. The first initial coupon of 2 presumably can be bought for 1.969 to yield 3.15% (s.a.). Then assume that the second coupon of 2 can be bought for 1.932 to yield 3.463% (s.a.). The third coupon of 2 can be bought for 1.891 to yield 3.788% (s.a.). The final cash flow of 102 must have been purchased for 94.208 if the strategy were to break even [= 100 - 1.969 - 1.932 - 1.891 = 94.208]. Hence, the 2.0year implied zero-coupon rate must be 4.013% (s.a.). In general, if the yield curve on coupon bonds (i.e., the par curve) is upward-sloping, the implied zero curve will lie above it. If the par curve is downward-sloping, the implied zero curve will lie below.
3 - 14 Implied Spot Yield Curves (cont.) Yield (s.a.) Implied Zero Curve 3.788% 3.463% 4.013% 4.00% 3.78% 3.46% 3.15% Par Curve 0.5 1.0 1.5 2.0 Maturity
The difference between the zero curve and the par curve (known as the coupon bias) depends on the level of rates, the steepness of the yield curve, and the time to maturity. 3 - 15 Implied Forward Rates Suppose that the yield curve for default free, zero-coupon bonds is as follows: Maturity Yield 1 2 3 4 5 7.0% 7.5% 8.0% 8.5% 9.0% If an investor wishes to make a two year investment, there are several different ways that this can be achieved. Two of these are: (i) Purchase a single two year bond, (ii) Purchase two consecutive one year bonds. With the first alternative, the investor is certain to have (1 + .075) 2 at the end of the two year holding period. With the second alternative, the investor is certain to have (1 + .07) at the end of the first one year holding period. 3 - 16
Implied Forward Rates (cont.) The question the investor must answer is: What is the rate expected to be on a one year investment one year from today? One way to answer this question is to calculate the yield one year from now that will allow the investor to break even between the two strategies. That is, calculate IFR1x2 such that: (1 + .075)2 = (1 + .07) x (1 + IFR1x2) or, 2 (1 + .075) -1 IFR1x2 = (1 + .07) = 8% The breakeven yield, IFR1x2, is called the implied forward rate of a one year security available one year in the future. If investors are pricing securities consistent with the unbiased expectations view, the implied forward rate also represents the market's prediction as to where one year interest rates will be one year forward. 3 - 17 Implied Forward Rates (cont.) For the yield curve listed above, the series of one year implied forward rates can be calculated as: Period IFR1x2 IFR2x3 IFR3x4 IFR4x5 Implied Forward Rate [(1.075)2(1.07)1]-1= 8% [(1.08)3(1.075)2]-1= 9%
[(1.085)4(1.08)3]-1=10% [(1.09)5(1.085)4]-1=11% Notice also that the five-year spot bond rate of 9% is simply the geometric average of the initial one-year spot rate and the four one-year implied forward rates. That is: 9% = 5 (1.07)(1.08)(1.09)(1.10)(1.11) - 1 In general, letting "A" represent the shorter maturity and "B" the longer one, this process can be illustrated: Year: 0 A B Actual Rate (B) Actual Rate (A) Implied Forward (B-A) 3 - 18 Implied Forward Rates (cont.) An important, sometimes overlooked, aspect to calculating implied forwards is that the cash market yields generally must have the same assumed number of periods in the year. Assume that the current (date 0) zero-coupon yields are denoted 0 x A and 0 x B. These are the annualized yields for compounding PER times per year, for A and B years to maturity, where B > A. With these assumptions, the generalized formula for calculating the implied forward yield between year A and year B is denoted A x B.
(Yield 0 x A ) 1 PER A Years x PER (Yield A X B ) x 1 PER (Yield 0 X B ) 1 PER (B - A years) x PER BYears x PER The terms in parentheses are one plus the periodic yields, that is, the annualized yields divided by the number of periods in the year. Note that the "PER" in the exponents cancels out from each side of the equation.
Example: What is the implied forward rate of a three-year bond available two years forward? Assume PER = 1 (1 +.09) IFR2x5 = 1 ( ) 5-2 5 (1 + .075) 2 -1 = 10% 3 - 19 Uses for Implied Forward Rates Predictions of Future Spot Rates: This assumes that investors set yield curves with unbiased expectations, which is seldom true. Generally, implied forward rates are upward-biased predictions of future spot rates because of liquidity premiums attached to yields of longer-term maturity bonds relative to shorter-term instruments Maturity Choice Decisions: Helps fixed-income investors decide on
appropriate maturity structure for a bond portfolio by quantifying the reinvestment rate embedded in longer-term securities compared to shorter-term ones Pricing Interest Rate Derivatives: Sets the arbitrage boundaries for the rates attached to actual forward agreements (e.g., bond futures, interest rate swaps) 3 - 20 U.S. Implied Forward Rates: February 2005 3 - 21 Basics of Bond Valuation Bonds are simply loans from bondholder to issuer (e.g., firm or government). Just like loans, bonds require interest payments and repayment of principal (also called face value or par) at a pre-specified future date. Interest payments are called coupon payments and bond principal repayments are usually non-amortizing (i.e., paid all at once at maturity) The current market value of a fixed-income bond is the present value of its future coupon and principal cash flows. In theory, the interest rates used to discount those future cash flows are the zero-coupon (or pure discount) rates corresponding to the dates of each cash flow. PMT PMT
PMT FV PV . . . 1 2 N N (1 z ) (1 z ) (1 z ) (1 z ) 1 2
N N 3 - 22 Basics of Bond Valuation (cont.) Consider a five-year, 9% (annual coupon payment) Eurobond. The market value of the bond, 103.99 (% of par value), can be obtained by calculating the present value of each scheduled cash flow using a sequence of zero-coupon rates commensurate with the riskiness of the bond. Period Cash Flow Zero-Coupon Rate 1 9 6.00% 8.49 2 9
7.00 7.86 3 9 7.50 7.24 4 9 7.80 6.66 5 109 8.13 73.74 Value = Present Value
103.99 3 - 23 Basics of Bond Valuation (cont.) The yield to maturity (y) of the bond is the constant interest rate per period that solves the following equation: PMT PMT PMT PV . . . 1 2 N (1 y ) (1 y ) (1 y
) FV (1 y) N The yield-to-maturity is the internal rate of return of all cash flows. It is the rate such that the present values of the cash flows, each discounted by that same rate, exactly equal the market value of the bond. The yield to maturity of this bond turns out to be 8.00%. 3 - 24 Basics of Bond Valuation (cont.) Period Cash Flow Yield-toMaturity 1 9 8.00% 8.33 2
9 8.00 7.72 3 9 8.00 7.14 4 9 8.00 6.62 5 109 8.00 74.18 Value =
Present Value 103.99 The yield to maturity is a statistic about the rate of return on the bond that includes both the coupon cash flows as well as any inevitable capital gain or loss if the bond is held to maturity (a gain if the bond is purchased at a discount below par value, a loss if the bond is purchased at a premium above par value). Therefore, it contains more information than the current yield, which is simply the coupon rate divided by the current price, e.g., 9 103.99 = .0865 . The current yield of 8.65% overstates the investors rate of return since it neglects the capital loss. 3 - 25 Basics of Bond Valuation (cont.) Notice that the yield to maturity can be interpreted as a "weighted average" of the sequence of zero-coupon rates, with most of the weight placed on the last cash flow since that is when the principal is redeemed, in that both deliver the same present value: 9 9 9 9 109
103.99 2 3 4 5 1.0600 (1.0700) (1.0750) (1.0780) (1.0813) 9 9 9 9 109 2 3 4 1.0800 (1.0800) (1.0800) (1.0800)
(1.0800)5 Clearly, the yield to maturity must lie within the range of the zerocoupon rates. 3 - 26 Current Coupon, Premium, and Discount Bonds A Current Coupon (or Par-Value) Bond is one for which the current market price equals the face value. In that case, the coupon rate (C/F) will equal the current yield (C/P), which will equal the yield-to-maturity (y). P=F <===> C/F = C/P = y The bond is priced at par value since its coupon rate is "fair" in that it equals the current market interest rate as represented by the yield-to-maturity. A Premium Bond has a current market price that exceeds the face value. In this case, the coupon rate will be higher than the current yield, which in turn will be higher than the yield-to-maturity. P>F <===> C/F > C/P > y The bond is priced at a premium above par value since its coupon rate is "high" given current market rates. A par-value, current coupon bond would have a lower coupon rate, so the premium represents the value of the "excessive" coupon cash flows. In fact, the amount of the premium is the present value of the annuity represented by the difference between the coupon rate and the bond's yield, discounted at that yield.
A Discount Bond has a current market price that is less than the face value. The coupon rate will be less than the current yield, which will be less than the yield-to-maturity. P
3.375 y 1 2 t 100 y 1 2 14 Solving for the periodic yield (i.e., y/2) on a financial calculator (such as the HP 12C) obtains 3.5472 [100 FV, 14 n, 3.375 PMT, -98.125 PV, i . 3.5472].
The annualized yield-to-maturity would then be reported as y = 7.0944% (i.e., 3.5472 x 2). 3 - 28 Interpreting Bond Information: Chile Govt 5.50% of January 2013 3 - 29 Interpreting Bond Information: ENDESA 8.35% of August 2013 3 - 30 Valuing the ENDESA 8.350% of 2013 Bond Between Coupon Dates | 2/1/05 | | | 2/18/05 8/1/05 8/1/13 n = 17 actual days (17 on 30/360 basis)
N = 181 actual days (180 on 30/360 basis) Bond Valuation: On February 1, 2005: P0 = 17 t =1 4.175 .0561315 1 2 t + 100 .0561315 1 2 17
= 118.3007 On February 18, 2005: Pn + AIn = (118.3007)(1 + 0.0561315/2)(17/180) = 118.6103 where: AIn = (4.175)(17/180) = 0.3943 so that: Clean or Flat Price = 118.6103 - 0.3943 = 118.2160 3 - 31 ENDESA 8.350% Bond of August 13 (cont.) 3 - 32 Sources of Bond Risk Primary: Default: Will the borrower honor its promise to repay? Interest Rate: How will changing market conditions affect the value of the bond?
Price risk component Reinvestment risk component Secondary: Call: Will the borrower refinance the loan under conditions that are disadvantageous to investor? Liquidity: How easily can bond be bought or sold? Tax: Will changes in the tax code affect bond values? 3 - 33 Bond Yields, Pricing, and Volatility Theorem #1: Bond prices are inversely related to bond yields. Implication: When market rates fall, bond prices rise, and vice versa. Theorem #2: Generally, for a given coupon rate, the longer is the term to maturity, the greater is the percentage price change for a given shift in yields. (The maturity effect) Implication: Long-term bonds are riskier than short-term bonds for a given shift in yields, but also have more potential for gain if rates fall. Theorem #3: For a given maturity, the lower is the coupon rate, the greater is the percentage price change for a given shift in yields. (The coupon effect)
Implication: Low-coupon bonds are riskier than high-coupon bonds given the same maturity, but also have more potential for gain if rates fall. Theorem #4: For a given coupon rate and maturity, the price increase from a given reduction in yield will always exceed the price decrease from an equivalent increase in yield. (The convexity effect) 3 - 34 Bond Yields, Pricing, and Volatility (cont.) Implication: There are potential gains from structuring a portfolio to be more convex (for a given yield and market value) since it will outperform a less convex portfolio in both a falling yield market as well as a rising yield Price Convex Price-Yield Curve Yield 3 - 35 Bond Yields, Pricing, and Volatility: Example Consider the following bonds: Bond Coupon
Maturity Initial Yield A 8% 5 yrs 10% B 5 20 8 C 8 20 8 Initial Prices: PA = PB =
PC = 5 80 1000 + = 924.18 t 5 (1 + .10) t =1 (1 + .10) 20 50 1000 + t (1 + .08) 20 t =1 (1 + .08) 20 80
1000 + t (1 + .08) 20 t =1 (1 + .08) = 705.46 = 1000.00 3 - 36 Bond Yields, Pricing, and Volatility: Example (cont.) Prices after yields increase by 50 bp: PA = PB = PC = 5 80
t t =1 (1 + .105) 20 50 t t =1 (1 + .085) 20 80 t t =1 (1 + .085) + +
+ 1000 (1 + .105) 5 1000 (1 + .085) 20 1000 (1 + .085) 20 = 906.43 = 668.78 = 952.68 Percentage price changes: Bond A: (906.43 - 924.18) / (924.18) = -1.92% (least) Bond B: (668.78 - 705.46) / (705.46) = -5.20% (most) Bond C: (952.68 - 1000.00) / (1000.00) = -4.73% (middle) 3 - 37 Bond Yields, Pricing, and Volatility: Example (cont.) Question: Where would Bond D, which has a coupon rate of 6% and a maturity of 19 years, fit into this price sensitivity spectrum? (Assume its initial yield is also 8%.) Initial: PD =
19 60 19 60 1000 + t (1 + .08)19 t =1 (1 + .08) = 807.93 After: PD = 1000 + t (1 + .085)19 t =1 (1 + .085) = 768.31
So, percentage change: Bond D: (768.31 - 807.93) / (807.93) = -4.90% 3 - 38 Motivating the Concept of Bond Duration In general, from the bond pricing theorems, the longer the term-to-maturity and the lower the coupon rate, the greater the percentage price change for a given change in yield. This suggests that many bonds can be ranked according to price volatility simply on the basis of their coupon rates and maturity dates, that is, by the "name" of the bond. Identifying Price Risk by Inspection of the Name of the Bond (i.e. its Coupon Rate and Maturity Date) Coupon NW Rate 12% NE * * * * * * * * ** ** * * * * *** * * ** * * * *** * * * ** * * * * * *
* * * * * ** * * * **** ** * **** * * * * * * ** ** * * * * * * * * * * * ** * *** * * *** * ** * * * * * *** ** *** * * SE SW * * * *** * * 5 Years Maturity The duration statistic will provide a one-to-one mapping between observable characteristics of a bond (its coupon rate, maturity, and yield) and its price risk arising from a volatile yield curve. 3 - 39 Motivating the Concept of Bond Duration (cont.) The key point is that more than one bond will "map" into the same point on the
duration scale. Assuming each bond has a yield to-maturity of 6% Coupon Rate 15.71% coupon, 7.0-year bond 11.25% coupon * 6.5-year bond 7.32% coupon * 6.0-year bond 3.63% coupon 5.5-year bond 0% coupon 5.0-year bond * * * Maturity 5 Duration So, at a first order of approximation each of these five bonds will experience the same percentage price change if the yield curve were to shift up or down in a parallel
manner, i.e., a uniform (or shape-preserving) shift. The bonds are price-risk equivalents. That's because they have the same duration statistic despite having different coupon rates and times to maturity. Also, each of these bonds would experience about half the percentage price change as would bonds that have a duration statistic of 10 and twice the change as bonds having a duration of 2.5. Duration can be used as a relative measure of price risk. 3 - 40 Calculating the Duration Statistic The duration of a bond is a weighted average of the payment dates, using the present value of the relative cash payments as the weights: PMT PMT 1 2 (1 y)
2 x (1 y) DUR 1 x PV0 PV0 PMT FV N (1 y)
... N x PV0 This statistic is the Macaulay duration, named after Frederick Macaulay who first developed it, and can be interpreted as the point in the life of the bond when the average cash flow is paid. 3 - 41 Calculating the Duration Statistic: Example or: Consider a five-year, 12% annual payment bond having a face value of $1,000. Suppose that the bond is priced at a premium to yield 10% (p.a.). The price of the bond is $1,075.82 and the Macaulay duration is
4.074: Year Cash Flow PV at 10% PV/Price Yr x (PV/Pr) 1 120 109.09 0.1014 0.1014 2 120 99.17 0.0922 0.1844 3
120 90.16 0.0838 0.2514 4 120 81.96 0.0762 0.3047 5 1120 695.43 0.6464 3.2321 $1075.82 1.0000 4.074 yrs
(109.09) 99.17 90.16 DUR 0 1 x 2 x 3 x 1,075.82 1,075.82 1,075.82 81.96 695.43
4 x 5 x 1,075.82 4.0740 1,075.82 3 - 42 Calculating the Duration Statistic: Closed-Form Equation The general formula for Macaulay duration can be reduced to a closed-form solution by taking the sums of the geometric series and algebraically rearranging the terms. 1+Y n D = Y n C Y 1 + Y n + (n x T) F - n C 1+YnxT - 1 + Y n n F where n is the number of payments per year, Y/n is the period yield, C/F is the coupon rate per period, and T is the years to maturity. Note that duration is calculated on the basis of the underlying periodic cash flows, even though it is typically is annualized when reporting the statistic
by dividing by n. In the preceding numerical example, Y = .10, n = 1, T = 5, C/F = .12, and Y/n = .10. The bond's duration can be solved as: 1 + .10 + 5 (.12 - .10) D = 1 + .10 = 4.0740 .10 5 .12 1 + .10 - 1 + .10 3 - 43 Duration as a Measure of Price Volatility Basic Price-Yield Elasticity Relationship: MV MV D (1 y/n) (1 y/n) Convert to Volatility Prediction Equation:
(1 y/n) y MV - (D) - (D) when n 1 MV (1 y/n) (1 y) Prediction Equation in Modified Form (% price change): D MV - (y) - (Mod D)(y) MV (1 y) 3 - 44
Duration as a Measure of Price Volatility (cont.) Convert to dollar (or cash) sensitivity: MV ~ -(Mod D)( y)(MV) Sensitivity to a one bp yield change (i.e., y = 0.0001): MV ~ -(Mod D)(0.0001)(MV) = Basis Point Value = BPV 3 - 45 Duration as a Measure of Price Volatility (cont.) Duration is a conservatively biased estimate of actual price changes from the perspective of the asset-holder. Duration underestimates price increases when rates fall and overestimates price reductions when rates rise. The reason for the bias is the convexity of the price-yield curve for a fixed-income instrument that contains no embedded put or call options. The bias is greater, the greater is the change in yield. For small rate changes, duration can be a very accurate predictor of the price change. Price Underestimate of Price Increase When the Yield Falls Overestimate of Price Decrease When the Yield Rises Convex Price-Yield Curve Yield 3 - 46
Duration and Price Volatility: Example Consider again the five-year, 12% coupon bond with a yield to maturity of 10%: Macaulay D: 4.074 Modified D: 3.704 (= 4.074 / 1.1) This means that an increase in yields of 100 bp will change the bonds price by approximately 3.7% in opposite direction Basis Point Value: $0.0398 [= (3.704)(.0001)(107.582)] This means that a one bp change in yields will cause the bonds price to move by about 4 cents per $100 of par value (which would correspond to a 40 cent movement for a bond with a par value of $1000) 3 - 47 Duration Example: ENDESA 8.350% of 2013 3 - 48
Duration of a Bond Portfolio The duration of a portfolio can be estimated as a weighted average of the durations of the securities in the portfolio, whereby the shares of total market value are the weights. Consider the following portfolio of semiannual payment bonds. Face Value Coupon Rate Maturity Yield (s.a.) Market Value $50 Million 8% 12 yrs 9.62% $44,306,787 $25 Million 11% 8 yrs 9.38% $27,243,887 $25 Million 9% 6 yrs 9.10% $24,886,343 $96,437,017 Duration 7.652 5.633 4.761 The duration of the portfolio can be estimated as: 44,306,787 27,243,887 24,886,343
7.652 x 5.633 x 4.761 x 96,437,017 96,437,017 96,437,017 = 6.336 3 - 49 LVACL Corporate Bond Index: Description & Performance 3 - 50 Example of Portfolio Duration: LVACL Corporate Bond Index 3 - 51 Example of Portfolio Duration: MBA Investment Fund Endowment Portfolio 3 - 52 Bond Convexity: An Overview On average, other things being equal, a more convex portfolio will outperform a less convex portfolio in both bull markets (falling rates, rising prices) and bear markets (rising rates, falling prices). This suggests that there might be gains from convexity that can be exploited in asset allocation. In general, this could
mean holding a "barbell" or "laddered" portfolio of diverse maturities rather than a "bullet" portfolio of concentrated maturities. Ladder Barbell Bullet For example, consider two portfolios with an equivalent duration--one, for example, a zero-coupon bond, and the other, a collection of coupon bonds. The coupon bonds will be more convex. Price Two Portfolios That Have the Same Yield, Same Market Value, and the Same Duration but Different Degrees of Convexity More Convex Portfolio Less Convex Portfolio Yield 3 - 53 Using Bond Convexity in Estimating Price Volatility Consider the 8%, semiannual coupon payment, 20-year bond that is priced at 110.678 to yield 7.00% (s.a.). The Macaulay duration is 10.760, the modified duration 10.396. The change in market value (MV) can be estimated as the product of the modified duration and the market value (i.e., the money duration) and the change in the yield to maturity (YLD):
MV - (Modified Duration x MV) x YLD But that linear estimator works best with small changes in the yield. For larger changes, the inherent convexity in the market-value-to-yield relationship leads to large errors. The estimate can be much improved by including the convexity statistic: MV - (Modified Duration x MV) x YLD 1/2 x (Convexity x MV) x (YLD)2 3 - 54 Using Bond Convexity in Estimating Price Volatility (cont.) The convexity statistic on this 8%, 20-year bond yielding 7.00% is 158.548. Consider a drop in the yield to 6.00%. Duration alone estimates a change in market value of + 11.506, whereas the actual change would be +12.437, a difference of +0.931. Including the convexity statistic reduces that difference to only +0.054: MV [- (10.396 x 110.678) x (-.0100)] [1/2 x (158.548x 110.678) x (-.0100)2 ] = 11.506 + 0.8777 = 12.383 3 - 55 Convexity Trades: An Example (Source: R. Dattareya and F. Fabbozi) Consider the following hypothetical U.S. Treasury bonds: Bond Coupon %
Maturity (yrs) Invoice Price Yield % Dollar Duration Dollar Convexity A 8.50 5 100 8.50 4.005 19.8164 B 9.50 20
100 9.50 8.882 124.1702 C 9.25 10 100 9.25 6.434 55.4506 Consider two different bond portfolios: Bullet Portfolio: 100% of Bond C Barbell Portfolio: 50.2% of Bond A, 49.8% of Bond B Notice the following:
Duration of Barbell: (.502)(4.005)+(.498)(8.882) = 6.434 Same as Bullet Portfolio Convexity of Barbell: (.502)(19.82)+(.498)(124.17) = 71.7846 Greater than Bullet Portfolio 3 - 56 Relative Performance (Bullet Rtn Barbell Rtn) Over Six-Month Period 3 - 57 Embedded Bond Options and Negative Convexity A callable bond allows the issuer to buy the bond back from the investor at a preagreed price (the call price) on certain dates (call dates). The call price on corporate bonds is typically a premium above par that declines as maturity nears. Treasuries, however, when callable are callable at par. Similarly, homeowners usually can prepay their mortgages without penalty. The call option will be exercised if the issuer in able to refinance the debt at a lower cost of funds due to lower market rates in general (a lower benchmark Treasury) or to an improvement in the issuers credit quality (a lower spread over the benchmark Treasury). Naturally the bondholder suffers when the call option is exercised because the proceeds would have to be reinvested at a lower rate or in a less creditworthy firm. The constraint on upside price appreciation as yields fall is known as negative convexity of the price-yield relationship: Price
Convex Price-Yield Relationship on a Bond with No Embedded Options Negative Convexity on a Callable Bond Yield 3 - 58 Embedded Bond Options and Negative Convexity (cont.) The presence of call risk to the investor requires an increase in the yield on a callable bond depending on the design of the option: The higher the call price, the lower the yield. The longer the call protection period, the lower the yield. From the perspective of the investor, buying a callable bond is equivalent to buying an otherwise comparable noncallable and writing a call option. + Callable bond = + Noncallable bond - Call option whereby "+" indicates a long position and "-" a short position. The higher yield received on the callable corresponds to the amortized value of the premium on the embedded option that has been written. Issuing a callable bond is equivalent to issuing a noncallable and buying a call option. - Callable bond = - Noncallable bond + Call option Similarly, the higher cost of funds on issuing the callable corresponds to the amortized value of the option (to refinance if yields fall) that has been purchased. The yield statistic on a callable bond is complicated by the presence of the option.
One simple remedy is to calculate the yield to each call date, using the call price that would apply to the redemption amount for that date. The lowest of the yields-to-call is referred to as the yield-to-worst. 3 - 59 Callable Bond Example: SBC 6.28% of October 2010-04 3 - 60 Callable Bond Example: SBC 6.28% of October 2010-04 3 - 61 Callable Bond Example: ENTEL 7.00% of January 2010-04 3 - 62 Overview of Bond Portfolio Strategies Generally speaking, bond portfolio management strategies can be broken down into four groups: I. Passive portfolio strategies a. Buy and Hold: This involves selecting securities with the desired characteristics (e.g., credit quality, coupon, maturity) and then not trading them again until they mature. b. Indexing: This is an attempt to design a portfolio of bonds that mimics a certain index (e.g., Lehman Brothers Index). There are two approaches to indexing: (1) full replication, in which all of the securities represented in the index are held in their exact proportions, and (2) stratified sampling, which divides the index into cells based upon parameters such as coupon, maturity, country, etc. and concentrates on indexing within certain cells. 3 - 63
Overview of Bond Portfolio Strategies (cont.) 3 - 64 Overview of Bond Portfolio Strategies (cont.) II. Active management strategies The active approach to bond portfolio management involves altering the portfolio so that the characteristics of the securities held provide the best chance of taking advantage of the managers view of anticipated conditions in the macro economy or with the specific security. Generally, the following types of views are employed in practice: a. Interest Rate Anticipation: This involves trading in and out of existing securities based on uncertain forecasts of future interest rate conditions; for this reason it is considered to be a risky strategy. b. Valuation Analysis: An attempt to select bonds based on their intrinsic value by accurately gauging the cost of the relevant characteristics (e.g., credit quality) of the bond. c. Credit Analysis: This involves assessing the default risk of an issuer and trading the bond when your perception differs from that of the market. 3 - 65 Overview of Bond Portfolio Strategies (cont.) II. Active management strategies (cont.) d. Yield Spread Analysis: A variation of interest rate anticipation, this strategy attempts to predict when the credit spread for a certain credit class will widen or narrow in anticipation of changing economic conditions. e. Bond Swaps: This involves liquidating a current position and simultaneously buying a different issue in its place with similar attributes, but a chance for improved return. Notable examples of this approach include pure yield pickup swaps, substitution swaps, and tax swaps. While these transactions can involve anything from little risk (yield pickup swaps) to great risk (rate anticipation swaps), they all are based on the premise that it is possible to improve portfolio performance as a result of
changing market conditions. 3 - 66 Overview of Bond Portfolio Strategies (cont.) II. Active management strategies (cont.) Ronald Layard-Liesching of Pareto Partners assess the likelihood of adding alpha to a fixed-income portfolio with active strategies as follows: Source Duration Yield Curve Sector Allocation Country Allocation Security Selection Optionality Prepayment Anomaly Capture Credit Risk Liquidity Currency Scale High Low High High Low Medium Medium Low High Low High
Sustainable Very Weak Very Weak Strong Strong Medium Medium Medium Weak Strong Strong Medium Information Ratio* 1 3 6 5 5 7 6 7 8 3 2 Extreme Values Yes No No
No No Yes Yes Yes Yes Yes Yes * Range is from 1 (low) to 10 (high) 3 - 67 Examples of Typical Yield Curve Shifts 3 - 68 Active Bond Trades: Examples For each of the following bond trades, identify the reason(s) investors may have had in making each swap. [Note: Assume that each swap was executed in 1982.] Action Call Price YTM (a) Sell: Baa-1 Georgia Pwr 1st Mtg. 11.625% due 2000 108.24
75.625 15.71% Buy: Baa-1 Georgia Pwr 1st Mtg. 7.325% due 2001 105.20 51.125 15.39% (b) Sell: Aaa Amer. Tel. & Tel. Notes 13.25% due 1991 101.50 96.125 14.02% NC 102.200 13.83% NC 25.250 14.37%
Buy: Aa-1 Chase Manhattan Notes Floating Rate due 2009 103.90 90.250 --- (d) Sell: A-1 Texas Oil & Gas 1st Mtg. 8.25% due 1997 105.75 60.000 15.09% NC 65.600 12.98% 103.90 62.750 10.83% 109.86
73.000 16.26% Buy: U.S. Treasury Notes 14.25% due 1991 (c) Sell: Aa-1 Chase Manhattan Notes 0% due 1992 Buy: U.S. Treasury Bond 8.25% due 2005 (e) Sell: A-1 K Mart Convertible Deb. 6% due 1999 Buy: A-2 Lucky Stores S.F. Deb. 11.75% due 2005 3 - 69 Bond Swaps Another type of active trade is a bond swap. This involves liquidating a current position and simultaneously buying a different issue in its place with similar attributes, but a chance of improved returns. Notable examples of bond swaps include:
Pure Yield Pickup Swaps: Swapping out of a low-coupon bond into a comparable higher-coupon bond to realize an automatic and instantaneous increase in current yield and yield to maturity. Substitution Swaps: Swapping comparable bonds that are trading at different yields; based on the premise that the credit market is temporarily out of balance. Tax Swaps: Trades motivated by prevailing tax codes and accumulated capital gains in a portfolio (e.g., selling a bond with a capital loss to offset one with a capital gain). 3 - 70 Bond Swap Example Evaluate the following pure yield pickup swap: You are currently holding a 20-year, Aa-rated, 9.0 percent coupon bond priced to yield 11.0 percent. As a swap candidate, you are considering a 20-year, Aarated, 11.0 percent coupon bond priced to yield 11.5 percent You can assume that all cash flows are reinvested at 11.5 percent. 3 - 71 Bond Swap Example: Solution
3 - 72 Overview of Bond Portfolio Strategies (cont.) III. Matched Funding Techniques Generally, matched funding techniques are asset/liability management procedures, whereby the bond portfolio is managed with respect to a specific set of future liabilities. a. Cash Matching/Dedicated: This approach designing a bond portfolio so that it delivers cash in the exact amount and timing as it is needed to pay off a set of liabilities. This can be done strictly or with reinvestment of excess cash flows from prior periods to enhance returns. b. Duration Matching/Classical Immunization: This strategy involves holding a portfolio whose duration is equivalent to the duration of the underlying liabilities. In this manner, it is possible to eliminate interest rate risk as that exposures two componentsprice and reinvestment riskwill offset each other. More detailed notes on this process are given on the following pages. c. Horizon Matching: This is a combination of cash-matching dedication and immunization, based on a division of the liability cash flows into two segments involve short- and long-term time horizons. 3 - 73 3 - 74 The Mechanics of Bond Immunization Suppose that an investor wants to invest money in the bond market for a period of three years. The interest rate is currently 10% but it falls to 8% as soon as the initial investment is made. What is the actual rate of return that this investor enjoys in each of the following scenarios? (a) Purchase a 10-year bond paying a 9% annual coupon and sell it in three years; (b) Purchase three 1-year "pure discount" (i.e., zero-coupon) bonds; (c) Purchase a 3-year, pure discount bond; (d) Purchase a 4-year bond paying a 34.85% annual coupon and sell it in three
years. Bond A: Initial Investment: 10 P0 = 90 1000 t + .10) (1 + .10)10 (1 + t =1 = $938.55 Terminal (i.e., Year 3) Value: (i) Sale of Bond: 7 P3 = 90 (1 + .08) t= 1 t +
1000 (1 + .08)7 = $1052.06 (ii) Coupon Payments: 90(1 + .08)2 + 90(1 + .08) + 90 = $292.18 so, Total Terminal Value = $1344.24 and, thus, the annual realized rate of return is: 1344.24 ARR = 3 - 1 = 12.72% 938.55 3 - 75 The Mechanics of Bond Immunization (cont.) Bond B: Invest $1000 initially at 10% and reinvest total proceeds for two more years at 8%: Year 1: (1000.00)(1 + .10) = $1100.00 Year 2: (1100.00)(1 + .08) = $1188.00 Year 3: (1188.00)(1 + .08) = $1283.04 so, the realized rate of return is: 1283.04 ARR = 3 1000 - 1 = 8.66% Bond C: Current Price (for $1000 face value): P0 = (1000) (1 + .10)3 = $751.31 so, the realized rate of return is: 1000.00 ARR = 3 - 1 = 10.00%
751.31 Bond D: Initial Investment: 4 P0 = 348.50 1000 + + .10)t (1 + .10)4 (1 t =1 = $1787.71 Terminal (i.e., Year 3) Value: (i) Sale of Bond: P3 = (1000 + 348.50) (1 + .08) = $1248.61 (ii) Coupon Payments: 348.50(1 + .08)2 + 348.50(1 + .08) + 348.50 = $1131.37 so, Total Terminal Value = $2379.98 and, thus, the realized rate of return is: 2379.98 ARR = 3 - 1 = 10.00% 1787.71 3 - 76 The Mechanics of Bond Immunization (cont.) Notice that only for Bonds C and D does the yield to maturity (i.e., the expected return) equal the
realized rate of return (ARR) over the three investment horizon. To see why this is the case, consider the duration of each of these bonds: Bond A: Bond B: Bond C: Bond D: 6.89 years 1.00 year (per bond) 3.00 years 3.00 years Because the investor's planning horizon was three years, the only bonds that actually produced the expected yield of 10% were the two that had a duration of three years. Put another way, by investing in a bond that pays out the "average" cash flow at precisely the time it is desired, it is possible to completely offset the effects of a subsequent change in interest rates. If interest rates fall (rise) after purchase, the bond price will rise (fall) by exactly enough to offset the decline (increase) in income from reinvested coupons. This selection process is known as immunization. Given that bond risk caused by changing interest rates can be dichotomized into price risk and reinvestment risk, the following general statements can be made: If Duration > Planning Horizon, the investor faces Net Price Risk (i.e., Bond A) If Duration < Planning Horizon, the investor faces Net Reinvestment Risk (i.e., Bond B) If Duration = Planning Horizon, the investor is immunized (i.e., Bonds C and D) 3 - 77 Overview of Bond Portfolio Strategies (cont.) IV. Derivative-Linked & Contingent Procedures Contingent management procedures are hybrid strategies that attempt to marry the best practices from both passive and active strategies with the risk control mechanisms implied by dedicated strategies. a. Contingent Immunization: This approach combines classical immunization and
active management by accepting a lower lock in yield than prevails in the market on the immunized portion of the portfolio in exchange for some potential upside through active strategies. b. Enhanced Indexing: Also called indexing plus, this strategy supplements an indexed position with enough active position bets. An enhanced indexing strategy hopes to exceed the total return performance of the managers designated benchmark on a enough consistent basis to cover the costs of active management. Under this strategy, the total return on the index becomes the minimum objective rather than the target itself. 3 - 78 Overview of Bond Portfolio Strategies (cont.) 3 - 79 Overview of Bond Portfolio Strategies (cont.) IV. Derivative-Linked & Contingent Procedures (cont.) c. Core-Plus Management: Similar to enhanced indexing, a core-plus mandate attempts to add performance alpha by supplementing a fairly passive approach to managing the main fixed-income assets (i.e., the core) relative to the designated benchmark with plus investing in sectors not covered in the benchmark. A typical core portfolio might include government, agency, and investment-grade sectors, with the plus sectors then including high-yield and emerging market bonds. d. Derivative-Linked Investing: Derivativessuch as futures contracts, swap agreements, or optionsare used to transform the cash flows and/or risk profile of a plain vanilla bond offering it something other than the original structure. Although derivatives represent a cost-effective way of implement this sort of financial reengineering, the point using derivatives in the fixed-income portfolio is usually to
create something synthetically that: (i) the manager cannot do directly, or (ii) does not exist otherwise. 3 - 80 Overview of Bond Portfolio Strategies (cont.) 3 - 81 An Overview of Equity Alternatives As we have seen, debt and equity securities are the fundamental cornerstones of the capital markets. They represent the most prevalent securities that companies use to raise external funds and that investors purchase to hold in their portfolios. Often, however, there will be cases when either investors or issuers will want to do a transaction involving securities with an equity-like payoff structure, but they may choose not (or otherwise be unable) to use plain vanilla equity directly. Some reasons why conventional stock shares may not be appropriate even when an equity payoff is desired include: A corporation seeking to raise additional capital may find the market for its common stock to be unreceptive, perhaps due to other recent issuances. An institutional investor may be restricted from holding equity directly but can purchase a debt instrument with a equity-like principal payoff at maturity. A company may be able to lower the present cost of a debt financing by structuring a bond contract that allows investors the right to convert the debt into common equity at a future date.
We will look at two alternative forms of equity along these lines: (i) convertible securities, and (ii) structured notes 3 - 82 Notion of Convertible Bonds A convertible bond can be viewed as a pre-packaged portfolio containing two distinct securities: (i) a regular bond and (i) an option to exchange the bond for a pre-specified number of shares of the issuing firms common stock. Thus, a convertible bond represents a hybrid investment involving elements of both the debt and equity markets. The option involved can be viewed as either a put (i.e., the investor has the right to sell the bond back to the issuer and receive a fixed number of shares) or a call (i.e., the investor can buy a fixed number of shares from the issuing company, paid for with the bond). From the investors standpoint, there are both advantages and disadvantages to this packaging. Specifically, while buyer receives equitylike returns with a guaranteed terminal payoff equal to the bonds face value, he or she must also pay the option premium, which is embedded in the price of the security. Conversely, the issuer of a convertible bond increases the companys leverage while providing a potential source of equity financing in the future.
This arrangement may be particularly useful as a means for low-rated issuers to borrow money more cheaply in the present than with a straight debt issue while creating a potential demand for their shares if future conditions are favorable. 3 - 83 Convertible Bond Example: Cypress Semiconductor As an example of how one such issue is structured and priced, consider the 4.00 percent coupon convertible subordinated notes (sub cv nt) maturing in February of 2005 issued by a NYSE-traded company, Cypress Semiconductor Corporation (CY). Cypress Semiconductor designs, develops, manufactures and markets a broad line of high-performance digital and mixed-signal integrated circuits for a range of markets, including data communications, telecommunications, computers and instrumentation systems. The Bloomberg screen on the next slide shows the issues CUSIP identifier, contract terms and default rating, (i.e., B1), and indicates that this bond pays interest semi-annually on February 1 and August 1. The bond issue has $283 million outstanding and is callable at 101 percent of par. At the time of this report (i.e. February 2001), the listed price of the convertible was 92 percent of par and the price of Cypress Semiconductor common stock was 27.375 per share. 3 - 84
CY Convertible Bond Example (cont.) 3 - 85 CY Convertible Bond Example (cont.) As spelled out at the top of this display, each $1,000 face value of this bond can be converted into 21.6216 shares of Cypress Semiconductor common stock. This statistic is called the instruments conversion ratio. At the current share price of $27.375, an investor exercising her conversion option would have received only $591.89 (= $27.375 21.6216) worth of stock, an amount considerably below the current market value of the bond. In fact, the conversion parity price (i.e. the common stock price at which immediate conversion would make sense) is equal to $42.55, which is the bond price of $920 divided by the conversion ratio of 21.6216. The prevailing market price of 27.375 is far below this parity level, meaning that the conversion option is currently out of the money. Of course, if the conversion parity price ever fell below the market price for the common stock, an astute investor could buy the bond and immediately exchange it into stock with a greater market value. 3 - 86 CY Convertible Bond Example (cont.) Most convertible bonds are also callable by the issuer. Of course, a firm will never call a bond selling for less than its call price (which is
the case with the Cypress Semiconductor note). In fact, firms often wait until the bond is selling for significantly more than its call price before calling it. If the company calls the bond under these conditions, investors will have an incentive to convert the bond into the stock that is worth more than they would receive from the call price; this situation is referred to as forcing conversion. Two other factors also increase the investors incentive to convert their bonds. First, some instruments have conversion prices that step up over time according to a predetermined schedule. Since a stepped up conversion price leads to a lower number of shares received, it becomes more likely that investors will exercise their option just before the conversion price increases. Second, a firm can help to encourage conversion by increasing the dividends on the stock, thereby making the income generated by the shares more attractive relative to the income from the bond. 3 - 87 CY Convertible Bond Example (cont.) Another important characteristic when evaluating convertible bonds is the payback or break-even time, which measures how long the higher interest income from the convertible bond (compared to the dividend income from the common stock) must persist to make up for the difference between the price of the bond and its conversion value (i.e., the conversion premium). The calculation is as follows: Payback =
Bond Price Conversion Value . Bond Income Income from Equal Investment in Common Stock For instance, the annual coupon yield payment on the Cypress Semiconductor convertible bond is $40, while the firms dividend yield is zero. Thus, assuming you sold the bond for 920 and used the proceeds to purchase 33.607 shares (= $920/$27.375) of Cypress Semiconductor stock, the payback period would be: $920.00 $591.89 8.20 years. $40.00 $0.00 3 - 88 CY Convertible Bond Example (cont.) It is also possible to calculate the combined value of the investors conversion option and issuers call feature that are embedded in the note. In the Cypress Semiconductor example, with a market price of $920, the convertibles yield-tomaturity can be calculated as the solution to: 8 $920 t 1 20 1000 (1 y / 2) t (1 y / 2) 8 or y = 6.29 percent. This computation assumes 8 semi-annual coupon payments of
$20 (= 40 2). Since the yield on a Cypress Semiconductor debt issue with no embedded options and the same (B1) credit rating and maturity was 8.5 percent, the present value of a straight fixed-income security with the same cash flows would be: 8 $850.06 t 1 20 1000 (1 0.0425) t (1 0.0425) 8 This means that the net value of the combined options is $69.94, or $920 minus $850.06. Using the Black-Scholes valuation model, it is easily confirmed that a fouryear call option to buy one share of Cypress Semiconductor stock which does not pay a dividend at an exercise price of $42.55 (i.e. the conversion parity value) is equal to $6.35. Thus the value of the investors conversion option which allows for the acquisition of 21.6216 shares must be $137.26 (= 21.6216 $6.35). This means that the value of the issuers call feature under these conditions must be $67.32 (= $137.26 $69.94). 3 - 89 Illustrating Convertible Bond Valuation 3 - 90 Notion of Structured Notes Generally speaking, structured notes are debt issues that have their principal or coupon payments linked to some other underlying variable.
Examples would include bonds whose coupons are tied to the appreciation of an equity index such as the S&P 500 or a zero-coupon bond with a principal amount tied to the appreciation of an oil price index. There are several common features that distinguish structured notes from regular fixed-income securities, two of which are important for our discussion. First, structured notes are designed for (are targeted to) a specific investor with a very particular need. That is, these are not "generic" instruments, but products tailored to address an investor's special constraints, which are often themselves created by tax, regulatory, or institutional policy restrictions. Second, after structuring the financing to meet the investor's needs, the issuer will typically hedge that unique exposure with swaps or exchangetraded derivatives. Inasmuch as the structured note itself most likely required an embedded derivative to create the desired payoff structure for the investor, this unwinding of the derivative position by the issuer generates an additional source profit opportunity for the bond underwriter. 3 - 91 Overview of the Structured Note Market 3 - 92 Equity Index-Linked Note Example: MITTS In July of 1992, Merrill Lynch & Co. raised USD 77,500,000 by issuing 7,750,000 units of an S&P 500 Market Index Target-Term Security, or "MITTS" for short, at a price of USD 10 per unit. These
MITTS units had a maturity date of August 29, 1997, making them comparable in form to a five-year bond even though they traded on the New York Stock Exchange. Indeed, Merrill Lynch issued them as a series of Senior Debt Securities making no coupon payments prior to maturity. At maturity, a unit holder received the original issue price plus a "supplemental redemption amount," the value of which depended on where the Standard & Poor's 500 index settled relative to a predetermined initial level. Given that this supplemental amount could not be less than zero, the total payout to the investor at maturity can be written: Final S & P Value - Initial S& P Value x 1.15 Initial S & P Value 10 + Max 0, 10 x where the initial S&P value was specified as 412.08. 3 - 93
MITTS Example (cont.) From the preceding description, recognize that the MITTS structure combines a five-year, zero-coupon bond with an S&P index call option, both of which were issued by Merrill Lynch. Thus, the MITTS investor essentially owns a "portfolio" that is: (i) long in a bond and (ii) long in an index call option position. This particular security was designed primarily for those investors who wanted to participate in the equity market but, for regulatory or taxation reasons, were not permitted to do so directly. For example, the manager of a fixed-income mutual fund might be able to enhance her return performance by purchasing this "bond" and then hoping for an appreciating stock market. Notice that the use of the call option in this design makes it fairly easy for Merrill Lynch to market to its institutional customers in that it is a "no lose" proposition; the worst-case scenario for the investor in that she simply gets her money back without interest in five years. (Of course, the customer does carry the company's credit risk for this period.) Thus, at origination the MITTS issue had no downside exposure to stock price declines. 3 - 94 MITTS Example (cont.)
The call option embedded in this structure is actually a partial position. To see this, we can rewrite the option portion of the note's redemption value as: = Max Final S & P - 412. 08 x 1.15 11. 50 0 , Final S & P 412. 08 412. 08 Max 0, 10 x (0. 0279) Max 0, Final S & P
- 412. 08 - 412. 08. Thus, given that a regular index call option would have a terminal payoff of Max[0, (Final S&P X)], where X is the strike price, the derivative in the MITTS represents 2.79% of this amount. 3 - 95 MITTS Example (cont.) On February 28, 1996, the closing price for the MITTS issue was USD 15.625, while the S&P 500 closed at 644.75. Further, the semi-annually compounded yield of a zero-coupon (i.e., "stripped") Treasury bond on this date was 5.35%. Assuming a credit spread of 30 basis points to be appropriate for
Merrill Lynch's credit rating (i.e., A+ and A1 by Standard & Poor's and Moody's, respectively) and the remaining time to maturity (i.e., one-and-a-half years, or three half-years), the bond portion of the MITTS issue should be worth: MITTS Bond Value = 10 .0565 1 + 2 3 = 9.20. This means that the investor is paying $6.43 (= 15.63 - 9.20) for the embedded index call. 3 - 96 MITTS Example (cont.) Without reproducing the full calculations, it is interesting to note that the theoretical value on February 28, 1996 of an S&P index call option expiring on August 29, 1997 with an exercise price of 412.08 is $243.19.
Thus, since the MITTS option feature represents 0.0279 of this amount, the call option embedded in the MITTS issue is worth $6.78 (= 243.19 x 0.0279). Thus, on this particular date the MITTS issue was priced in the market below its theoretical value, presenting investors with a potential buying opportunity depending on their transaction costs. In fact, the embedded call is actually priced below the index options intrinsic value of $6.49 (= [644.75 412.08] x 0.0279), making the issue that much more attractive to investors. 3 - 97 MITTS Example (cont.) This MITTS transaction can be illustrated as follows: Max(0,% SPX Rtn) $10 August 1992 February 1996 August 1997 Zero-Coupon Bond $9.20 $10 SPX Index Call Option
$6.43 3 - 98 Additional Structured Note Examples 3 - 99 Additional Structured Note Examples (cont.) 3 - 100